You can use Web Entry to allow anonymous users to start cases in ProcessMaker. You can use the Case Tracker to allow anonymous users to access the information about particular cases, but you will need to send the user the case number and case PIN in order to be able to open a case in the Case Tracker. You can also use Actions By Email to send an email to the user and the user can fill out a form in the email.

If you want anonymous users to be able to do anything else (such as working on cases), then you will need to create a web site or script which uses REST or web services to access the cases. This will require some programming on your part.